ECE 663 So far, we looked at equilibrium charge distributions. The end result was np = n i 2 When the system is perturbed, the system tries to restore itself towards equilibrium through recombination-generation We will calculate the steady-state rates This rate will be proportional to the deviation from equilibrium, R = A(np-n i 2 ) R-G processes

ECE 663 h h Real spaceEnergy space Direct Band-to-band recombination Lasers, LEDs,..

ECE 663 h h Real spaceEnergy space Direct excitonic recombination Organic Solar cells, CNTs, wires (1-D systems)  

ECE 663 R-G for a Direct Band-gap Conserve momentum and Energy 1. E g = h 2.  k = k photon = 2  /c = E g /ħc Eg ~ 1.2 eV   k ~ 6/  m << BZ = 2  /a [Å] Optical transitions almost vertical !!

ECE 663 Indirect Band-gap Since photons make vertical transitions, they won’t conserve momentum for indirect band-gaps (Si, Ge)

ECE 663 Indirect Band-gap: What about phonons? Conserve momentum and Energy 1. E phonon = h phonon 2.  k = k phonon = 2  phonon /v sound = E phonon /ħv sound ≈ 2  /a [Å] Because they involve atomic Displacements, wavelengths are comparable But energy very small (~ meV). Not sufficient

ECE 663 Indirect Band-gap: Involving traps Two step process !! k trap ≈ 2  /a

ECE 663 Phonons Real spaceEnergy space Indirect (Trap-assisted) recombination Nonradiative Recombination (Ge, Si FETs, solar cells ) 

ECE 663 Shockley-Read-Hall

ECE 663 Phonon Real spaceEnergy space Auger Recombination III-Vs, highly doped samples (Opposite process is impact ionization) X X

ECE 663 Real spaceEnergy space Impact Ionization Si, Ge, InP Lasers, FETs X X

ECE 663 The capture process N T = n T + p T

ECE 663 The capture rate Assume these c’s don’t change under bias

ECE 663 Net Recombination Rates (1-f)

ECE 663 Net Recombination Rates

ECE 663 Problem Solving Strategy Look at equilibrium  e n Steady-state: Set r N = r P  n T Plug back in:  R = r N, r P Similar prescription in ECE 687 to get ballistic current, except R = A(n-f), not A(np-n i 2 )

ECE 663 Look at Equilibrium first Detailed Balance

ECE 663 Look at Equilibrium first r N = r P = 0 e n = c n p T0 n 0 /n T0 = c n n 1 e p = c p n T0 p 0 /p T0 = c p p 1 n 1 = n i e (E’ T -E i )/kT p 1 = n i e (E i -E’ T )/kT Charge densities if traps pin Fermi energy

ECE 663 Substituting:

ECE 663 Now look at steady-state

ECE 663 Detailed Balance, Steady State No net clockwise flow Steady clockwise flow

ECE 663 … and Transients Unsteady flow

ECE 663 Steady State

ECE 663 Steady State Recombination Rate R Use: And

ECE 663 Steady State Recombination Rate R n 1 = n i e (E’ T -E i )/kT p 1 = n i e (E i -E’ T )/kT c i =  i v th

Typical #s NTNT ETET  TN  TP SNSN SPSP Material [cm-3][eV][m2] [m/s] Si, Ge1e130.01e III-Vs2e160.41e-141e-130.0

ECE 663 Let’s look at a few limits n = n 0 +  n,  n << n 0 p = p 0 +  p,  p << p 0 << n 0 Low-level injection, n-type material  n ~  p Few traps Deep traps (midgap) n 1 ≈ p 1 ≈ n i

ECE 663 Low level injection Set n 0 p 0 = n i 2  cancels term in numerator Drop  n  p term in numerator n 0  p >> p 0  n in numerator for n-type material Keep only n 0 related term in denominator

ECE 663 For n-type: For p-type Low level injection We’ll frequently adopt this approximation

ECE 663 Surface Recombination Lattice periodicity broken at surface/interface – mid-gap E levels Carriers generated-recombined per unit area

ECE 663 Surface States Reconstruction Expt (Akiyama et al, PRB 2000) Theory (Rakshit/Liang/Ghosh/ Hersam/Datta, PRB 2005)

ECE 663 Processes and descriptions analogous to bulk R-G using surface parameters For a single energy level surface state: Surface Recombination

ECE 663 Surface Recombination CAVEAT !! s n and s p have units of length/time – surface recombination velocity for single level surface states

ECE 663 Multi-Level (more realistic) D IT (E) – density of interface traps (per unit area-energy) D IT (E)dE – density of IT between E and E+dE (replaces N Ts )

In summary R-G processes drive system towards equilibrium R  (np – n i 2 ) (In ECE687, restoration will be driven by the contacts) For indirect band-gap materials, SRH dominates Coeffs depend on minority carrier lifetime, a critical concept for this course Minority carrier lifetime depends on trap cross-section (size), trap density and electron thermal velocity When computing current, the drive forces (drift-diffusion) in the next chapter will be countered by these RG forces